Properties

Label 2-864-27.13-c1-0-9
Degree $2$
Conductor $864$
Sign $0.931 + 0.363i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.42 − 0.990i)3-s + (−2.13 − 1.79i)5-s + (2.82 + 1.02i)7-s + (1.03 + 2.81i)9-s + (0.488 − 0.409i)11-s + (−1.01 + 5.75i)13-s + (1.25 + 4.66i)15-s + (−0.343 + 0.595i)17-s + (0.768 + 1.33i)19-s + (−2.99 − 4.26i)21-s + (8.70 − 3.16i)23-s + (0.480 + 2.72i)25-s + (1.31 − 5.02i)27-s + (−0.568 − 3.22i)29-s + (−0.333 + 0.121i)31-s + ⋯
L(s)  = 1  + (−0.820 − 0.571i)3-s + (−0.954 − 0.801i)5-s + (1.06 + 0.389i)7-s + (0.345 + 0.938i)9-s + (0.147 − 0.123i)11-s + (−0.281 + 1.59i)13-s + (0.324 + 1.20i)15-s + (−0.0834 + 0.144i)17-s + (0.176 + 0.305i)19-s + (−0.654 − 0.931i)21-s + (1.81 − 0.660i)23-s + (0.0961 + 0.545i)25-s + (0.253 − 0.967i)27-s + (−0.105 − 0.598i)29-s + (−0.0599 + 0.0218i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.931 + 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07705 - 0.202879i\)
\(L(\frac12)\) \(\approx\) \(1.07705 - 0.202879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.42 + 0.990i)T \)
good5 \( 1 + (2.13 + 1.79i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2.82 - 1.02i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.488 + 0.409i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.01 - 5.75i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.343 - 0.595i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.768 - 1.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.70 + 3.16i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.568 + 3.22i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.333 - 0.121i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (3.90 - 6.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.503 + 2.85i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-8.93 + 7.49i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.06 - 0.752i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 9.65T + 53T^{2} \)
59 \( 1 + (-7.72 - 6.48i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-11.7 - 4.29i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.44 + 8.16i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (7.06 - 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.61 + 7.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0301 - 0.170i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.386 + 2.19i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (4.65 + 8.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.322 - 0.270i)T + (16.8 - 95.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.33140391841323804871953608176, −8.898470955127955895908334700672, −8.520069215179312900134557477758, −7.44893571709654377621884235644, −6.84597750757531722096908945487, −5.58920740723528721015967706568, −4.76392624531195669987612434002, −4.14887919196852776935859249059, −2.16707807537744477012077813069, −0.963506667694837169495208830988, 0.860298365277647575645433694411, 2.99298989612592557885836069557, 3.91107902086062905407471799209, 4.94490777843674333201015215604, 5.59426162330209643331624967676, 7.06562973161470559989396140821, 7.39845290721113718056340912352, 8.447413903693905180881069454958, 9.552446661846898275541874061717, 10.56999949069199640478367335482

Graph of the $Z$-function along the critical line